My take on this speach is that this is a proud man looking at his end-game who is trying to inspire thought, and maybe he's even looking for a protege. By the reaction I see here he has achieved at least one of those goals.

It is difficult to grasp one's imagination unless we become some part of it.
Do not know what to comment about this, but certainly we have become the most contradictory species on earth.
Nothing can help us here unless we learn how to maintain and sustain what we have.
For e.g., we speak of great innovations, but the products die at a faster rate than it took to build. Oh yeah, that is Moore's law..So how is it going to help.
Not to say that it might have made billions rendering worthless in few years. Yet, a lot of people complain about NASA's funding, for which there is no direct material return.
Imagine a day in 1500s or 1600s,can we survive it? Oh, that is a reality show..
Unless we practice science and respect it, no intuition or technically sophisticated mathematics would help. Are we simplifying or complicating? Time will tell for sure. The future looks bleak.

The fundamental problem with Quantum Physics is that is is an empirical science. We know that Schroedinger's equation works, but we have no concept of why. In that sense, the phrase "Quantum Theory" is a misnomer. There is no theory, only an equation that works. When there is an understanding of quantum behavior that Shroedinger's Equation can be derived from, we will have a theory. Until that theory is created, there are no intuitive concepts.

Whether Meade is right, I think depends on what he means. If he means that we need to find ways of communicating the concepts in an intuitive way, I agree with him. If he means that intuition is an important PART of FINDING the truth, I also agree with him. If he means that our current INTERPRETATION of the data and math needs to be challenged, I agree with that, too. If he means that rigor, analysis, and experimental validation are unneccessary, he is a fool.

The ability to understand that scientifically is a logical contradiction. Science is only able to deal with "repeatable phenomena". It therefore is only able to work with statistical averages of behavior. Trying to understand individuals or groups that cannot be directly influenced is beyond science's scope.

You can't go wrong by understanding as much math as possible. However, beware of becoming a mathematician.
There was a student-targeted publication I used to see when I was a kid, and its key motto emblazoned throughout each issue was "Chance favors the prepared mind". I would add that intuition, visualizations, even meditative activity can be spectacularly productive IF you have some good grounding in the physical and mathematical fundamentals.
It's not sufficient or necessary to acquire these latter via an advanced degree, but it makes it a whole lot easier, especially if you find some good teachers. The danger is getting blinkered and over-specialized, and also protective of your negative ego, which will devastate speculation and possible insight.
A while ago someone posted in a chatroom a nice problem, an apparent geometrical contradiction. When I told him what the solution was he said NO that's what everyone says. But my grad student math friend explained that it's about tessellations blah blah. I said your grad student friend panicked when he couldn't see the answer from elementary plane geometry, and proceeded to baffle you with BS. He should be ashamed.

You make it sound like intuition is somehow precedes actual knowledge and trumps tedious calculations. This is just incorrect: the great scientits who were famous for their intuition, like Newton, Einstein or Feynmann, were all brilliant technicians, and did the calculations so much they gave them intuition that transcended the conventional understanding.
In other words, their great intuitions were BUILT ON their technical prowess, not working against it. I am reminded of Salvatore Dali who is of course famous for surrealism but who was capable of super-realistic, photographic illustration.
It makes sense to me on a personal level that intuition and detailed grunge work are complementary and both need as well as enable each other. I intuitively know trigonometry because I did a lot of homework on it; now I have an intuitive understanding of sin()---and can leverage it to understand Bessel functions. My children perhaps will intuitively understand the behavior of J0().
This meme that intuition and gut feeling simply trumps reflection and rigor is very dangerous---unfortunately it insinuated itself into important areas such as politics and economy, and we'll suffer the ill effects of that for a long time. When Carver Mead asks us to question those who "know it's so" he requires a rigorous argument, not just another "I know it's otherwise".

"How many times has earlier science been later proven wrong?"
And exactly how does that happen?
Somebody models the results of a theory. Then he or she makes observations. Then he or she compares the observations to the model.
You can't do the modeling without the math. Intuition is no substitute.